Toeplitz determinants with merging singularities

Abstract

We study asymptotic behavior for the determinants of n×n Toeplitz matrices corresponding to symbols with two Fisher–Hartwig singularities at the distance 2t≥0 from each other on the unit circle. We obtain large n asymptotics which are uniform for 0<t<t0, where t0 is fixed. They describe the transition as t→0 between the asymptotic regimes of two singularities and one singularity. The asymptotics involve a particular solution to the Painlevé V equation. We obtain small and large argument expansions of this solution. As applications of our results, we prove a conjecture of Dyson on the largest occupation number in the ground state of a one-dimensional Bose gas, and a conjecture of Fyodorov and Keating on the second moment of powers of the characteristic polynomials of random matrices.

Locations

• Duke Mathematical Journal - View
• arXiv (Cornell University) - View - PDF
• DataCite API - View